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Variational methods in materials science

By A. Forclaz


Three problems are being investigated in this thesis. The first two relate to the modelling and analysis of martensitic phase transitions, while the third is concerned with some mathematical tools used in this setting. After a short introduction (Chapter 1) and overviews of the calculus of variations and martensitic phase transformations (Chapter 2), the research part of this thesis is divided into three chapters. We show in Chapter 3 that for the two wells $\mathrm{SO}(3)U$ and $\mathrm{SO}(3)V$ to be rank-one connected, where the $3\times 3$ symmetric positive definite $U$ and $V$ have the same eigenvalues, it is necessary and sufficient that $\mathrm{det}(U-V)=0$, a result that does not hold in higher dimensions. Using this criterion and a result of Gurtin, formulae for the twinning plane and the shearing vector are obtained, which yield an extremely simple condition for the occurrence of so-called compound twins. Our results also provide a simple classification of the twinning mode of the two wells by looking at the crystallographic properties of the eigenvectors of the difference $U-V$. As an illustration, we apply our results to cubic-to-tetra gonal,tetragonal-to-monoclinic and cubic-to-monoclinic transitions. Chapter 4 focuses on the mathematical analysis of biaxial loading experiments in martensite, more particularly on how hysteresis relates to metastability. These experiments were carried out by Chu and James and their mathematical treatment was initiated by Ball, Chu and James. Experimentally it is observed that a homogeneous deformation $y_1(x)= U_1x$ is the stable state for `small' loads while $y_2(x)=U_2x$ is stable for `large' loads. A model was proposed by Ball, Chu and James which, for a certain intermediate range of loads, predicts crucially that $y_1(x)=U_1x$ remains metastable i.e., a local - as opposed to global - minimiser of the energy). This result explains convincingly the hysteresis that is observed experimentally. It is easy to get an upper bound for when metastability finishes. However, it was also noticed that this bound (the Schmid Law) may not be sharp, though this required some geometric conditions on the sample. In this chapter, we rigorously justify the Ball-Chu-James model by means of De Giorgi's $\Gamma$-convergence, establish some properties of local minimisers of the (limiting) energy and prove the metastability result mentioned above. An important part of the chapter is then devoted to establishing which geometric conditions are necessary and sufficient for the counter-example to the Schmid Law to apply. Finally, Chapter 5 investigates the structure of the solutions to the two-well problem. Restricting ourselves to the subset $K=\{H\}\cup \mathrm{SO}(2)V \subset\mathrm{SO}(2)U\cup\mathrm{SO}(2)V$ and assuming the two wells to be compatible, we let $T_1$ and $T_2$ denote the two (not necessarily distinct) twins of $H$ on $\mathrm{SO}(2)V$ and ask the following question: if $\nu_x$ is a non-trivial gradient Young measure almost everywhere supported on $K$, does its support necessarily contain a pair of rank-one connected matrices on a set of positive measure? Although we do not provide a solution for the general case, we show that this is true whenever (a) $\nu_x\equiv \nu$ is homogeneous and $\mathrm{supp}\nu\cap \mathrm{SO}(2)V$ is connected, (b) $\nu_x\equiv \nu$ is homogeneous and $T_1=T_2$ i.e., when the two wells are trivially rank-one connected) or (c) $\mathrm{supp}\nu_x \subset F$ a.e., for some finite set $F$. We also establish a more general case provided a strong `rigidity' conjecture holds

Topics: Mechanics of deformable solids, Partial differential equations, Calculus of variations and optimal control
Year: 2002
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